I was given this $$M=\begin{pmatrix}-1 & x\\ -5 & y\end{pmatrix}$$ which has eigenvectors $$V=\begin{pmatrix}1\\ 1 \end{pmatrix}$$ and $$W=\begin{pmatrix}5\\ 1 \end{pmatrix}$$
I'm supposed to find the x and y along with the eigenvalues of both v and w but I don't really know how to start the solution. I just need a little kickstart I think, I think I'm fine with matrices but with eigenvectors put into it, I just can't get my head around it. Any help would be appreciated.
If $V$ is an eigenvector of $M$, that means there is a constant $\lambda_V$ such that $MV = \lambda_V V$. Plugging in $M$ and $V$ gives $$ \begin{pmatrix}x - 1 \\ y-5\end{pmatrix} = \begin{pmatrix}\lambda_V \\ \lambda_V\end{pmatrix} $$ So $x - 1 = \lambda_V$ and $y - 5 = \lambda_V$. You can do the same thing with $W$ to get two more equations for $x$, $y$, and $\lambda_W$. Now you have four linear equations in four unknowns, so you can solve for the answer.